L(s) = 1 | + 5-s − 2.60·7-s + 4.60·11-s − 6.60·13-s − 1.60·17-s − 3.60·19-s + 3·23-s + 25-s + 1.39·29-s − 5.60·31-s − 2.60·35-s − 2·37-s + 4.60·41-s − 0.605·43-s + 9.21·47-s − 0.211·49-s + 1.60·53-s + 4.60·55-s − 1.39·59-s + 4.21·61-s − 6.60·65-s + 0.788·67-s − 7.39·71-s + 12.6·73-s − 12·77-s − 11.6·79-s + 3·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.984·7-s + 1.38·11-s − 1.83·13-s − 0.389·17-s − 0.827·19-s + 0.625·23-s + 0.200·25-s + 0.258·29-s − 1.00·31-s − 0.440·35-s − 0.328·37-s + 0.719·41-s − 0.0923·43-s + 1.34·47-s − 0.0301·49-s + 0.220·53-s + 0.621·55-s − 0.181·59-s + 0.539·61-s − 0.819·65-s + 0.0963·67-s − 0.877·71-s + 1.47·73-s − 1.36·77-s − 1.30·79-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.551287605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.551287605\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 - 0.788T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53276786339395100095542627183, −7.00273798167659768202527728487, −6.48441555273343842544484916067, −5.82585689683415538050104793926, −4.95414213686917097731354119072, −4.26004200651020096661420244968, −3.46547036430954291079175300363, −2.58150263717831631719471914602, −1.88927133258516517891162305258, −0.58662922628036128644888636571,
0.58662922628036128644888636571, 1.88927133258516517891162305258, 2.58150263717831631719471914602, 3.46547036430954291079175300363, 4.26004200651020096661420244968, 4.95414213686917097731354119072, 5.82585689683415538050104793926, 6.48441555273343842544484916067, 7.00273798167659768202527728487, 7.53276786339395100095542627183